Naturalism & the Laws of Nature.

Theoretical physicist Paul Davies wrote:

But what are these ultimate laws and where do they come from? Such questions are often dismissed as being pointless or even unscientific. As the cosmologist Sean Carroll has written, “There is a chain of explanations concerning things that happen in the universe, which ultimately reaches to the fundamental laws of nature and stops… at the end of the day the laws are what they are… And that’s okay. I’m happy to take the universe just as we find it.”

Assuming that Davies is correct, I find it odd that there is little interest for understanding the laws of nature. There are some interesting questions to be answered, such as: Where do the laws come from? How do they cause things to happen?

Physicist Neil Turok once posed the question:

What is it that makes the electrons continue to follow the laws?

Indeed, what power compels physical objects to follow the laws of nature?

The question I would like to focus on is: what would a naturalistic explanation of the laws of nature look like?

Frankly, I don’t know where to start. What I do know is that a bottom-up explanation runs into a serious problem. A bottom-up explanation, from the level of say bosons, should be expected to give rise to innumerable different ever-changing laws. Different circumstances, different laws.

But this is not what we find. Again, Paul Davies:

Physical processes, however violent or complex, are thought to have absolutely no effect on the laws. There is thus a curious asymmetry: physical processes depend on laws but the laws do not depend on physical processes. Although this statement cannot be proved, it is widely accepted.

If laws do not depend on physical processes, then it follows that laws cannot be explained by physical processes. IOWs there is no bottom-up explanation for the laws of nature.

But what does it mean for naturalism if there is no bottom-up (naturalistic) explanation for the laws of nature? How does the central claim ‘everything is physical’ make sense if there is no physical explanation for the laws of nature? What if it is shown that the laws of nature control the physical but are not reducible to it?

 

 

 

364 thoughts on “Naturalism & the Laws of Nature.

  1. Erik,

    And you are still failing to see the contradiction.

    That’s because unlike you, I’m not a mathematical crackpot.

    Like other mathematically literate folks, I recognize that the empty set isn’t oxymoronic.

  2. Erik,

    Are you sticking with your crackpot claim that the empty set is not a set, and that the mathematical community has fallen into a trap that you — a mathematically unskilled linguist — have alone managed to avoid?

    Or has it finally dawned on you that that the mathematicians got it right, and that their notion of set is not only coherent, it also works better than yours?

  3. keiths: Are you sticking with your crackpot claim that the empty set is not a set…

    Did I claim this? If so, then you can quote me on this. We can go on once you get my position right.

  4. Erik:

    Did I claim this?

    Sure:

    Square circles are suspect, right? Similarly, empty sets are suspect. They involve definitional self-contradiction (supposing that a set is a collection of things) and this makes them suspect.

    Getting cold feet?

  5. keiths,

    Good. Now tell me how my claim that empty set is philosophically suspect equates with “empty set is not a set” as you put it. I doubt you will be able to cash this out. Just like Neil was unable to cash out his impression that I was claiming that 0=1.

  6. I have a better idea. Why don’t you tell us why you think the empty set involves “definitional self-contradiction” if you don’t claim that the empty set is not a set.

    Identify the “self-contradiction”.

  7. keiths: I have a better idea. Why don’t you tell us…

    Identify the “self-contradiction”.

    It’s in the quote that you quoted from me just one comment ago.

  8. You refer to a self-contradiction, but you don’t identify it.

    What is self-contradictory about the empty set if you are not claiming that the empty set is not a set?

    Stop evading the question.

  9. keiths: You refer to a self-contradiction, but you don’t identify it.

    Of course I did. Unfortunately it went over your head multiple times, while nobody else had any difficulty. Here it is once more just for you, for the last time.

    The relevant quote, “They involve definitional self-contradiction (supposing that a set is a collection of things) and this makes them suspect.” An empty set would have no things in it, so it would not be a collection of things. Hence the self-contradiction, like in square circle – a circle is round by definition, it cannot be square.

    Further, contrary to you, I am not claiming that empty sets are not sets. Neither am I claiming that square circles are not circles. Or not squares. And I am not claiming these are unusable concepts. They have their educational value, as a minimum. And given more nuanced definitions or elevated into status of axiom they may serve as a foundation of a viable-looking theory even. There are a bunch of scientific theories with philosophically suspect foundations. It’s nothing new, not the first time and not the last either.

    Once upon a time I encountered somewhere a wiseacre mathematician’s attempt to show that square circle is a normalcy. He asked to imagine a sphere, upon which we draw a square. His claim was that that would be something like a square circle. Now, this does show that precise definitions, context and presuppositions are important, but it doesn’t solve the fundamental self-contradiction, because the projection of a square on the surface of a sphere would not really be a square, just like the projection of a sphere on a plane is not a sphere. Thing versus its projection, different things. The guy in the mirror may very much look like you, but it isn’t you. To confuse the two is philosophically suspect.

    Square circle is a paradigm example of self-contradiction in philosophy and logic (as in “square peg in a round hole”). It doesn’t get any simpler than this.

  10. Erik,

    This is comical.

    First you tell us that the notion of the empty set is self-contradictory, because a set cannot be empty and still be a set:

    The relevant quote, “They involve definitional self-contradiction (supposing that a set is a collection of things) and this makes them suspect.” An empty set would have no things in it, so it would not be a collection of things.

    Then you turn around and tell us the opposite:

    Further, contrary to you, I am not claiming that empty sets are not sets.

    I know that disciplined thinking isn’t your forte, but surely even you can see that you’re contradicting yourself.

  11. keiths,

    Do you have any questions? No question marks in your latest post, so it looks like we finally settled the issue.

  12. This topic has been an embarrassment for you, so you’d clearly like to bring the discussion to an end, but no.

    We haven’t settled the issue unless you agree that

    a) the concept of the empty set is not self-contradictory;
    b) the mathematicians’ notion of set is coherent, and it works better than yours does; and
    c) you obviously contradicted yourself above as explained here.

    You embraced a crackpot notion: that the empty set is self-contradictory. That was a mistake, and you are clearly embarrassed by it, but it’s time to acknowledge the mistake, correct it, and move on.

  13. keiths: We haven’t settled the issue unless you agree that

    a) the concept of the empty set is not self-contradictory;

    I have explained how it is. You start explaining how it isn’t.

    keiths:
    b) the mathematicians’ notion of set is coherent, and it works better than yours does; and

    “Works better” for what purpose?

    keiths:
    c) you obviously contradicted yourself above as explained here.

    Your explanation reveals that you confuse the label “empty set” with the thing that would be denoted by it. “Empty set” is a self-contradictory label. Whatever is denoted by it, may be a set or not, or it may be empty or not. I’m not denying that mathematicians are talking about something noteworthy when they say “empty set”.

    Zero is mathematically a highly noteworthy concept; the issue is whether it’s justified to call it “empty set” which is a philosophically suspect label. The fundamental problem seems to be that you deny that “philosophically suspect” is something of importance.

    Experience shows that those who overlook self-contradictions are not worth talking to, particularly when they self-contradictorily cherry-pick some particular self-contradictions to overlook.

  14. keiths:

    This topic has been an embarrassment for you, so you’d clearly like to bring the discussion to an end, but no.

    We haven’t settled the issue unless you agree that

    a) the concept of the empty set is not self-contradictory;

    Erik:

    I have explained how it is. You start explaining how it isn’t.

    I’ve explained it again and again. The contradiction arises only for you, because only youinsist that sets must contain at least one element. Mathematicians are smarter than you. They adopt a more inclusive notion of set, under which there is nothing at all self-contradictory about the empty set.

    It’s an obvious solution, but you weren’t bright enough to see it.

  15. Erik,

    “Works better” for what purpose?

    For the purposes of set theory.

    Here’s an example. Using the mathematicians’ concept of “set”, the intersection of two sets A and B can be defined elegantly as

    A ∩ B = {x : x ∈ A ∧ x ∈ B}

    …but this won’t work with the clunky Erik notion of “set”.

  16. It doesn’t imply that intersections won’t work (or whatever it is you are trying to say there).

    The problem with the set theory is they so desperately want to call even non-sets sets (“empty sets”). What’s wrong with calling it zero or emptiness? It’s emptiness, nothing, so what’s the urge to call it a set?

    A∪∅=A ; Unite A with nothing, you got A
    A∩∅=∅ ; The intersection of A with nothing is nothing, as good as if there were no intersection
    and x∈∅ is always false. No elements in emptiness.

    I’m not saying that ∅ is a useless symbol. I’m saying that it’s a stretch to say that it represents a set.

    Moreover, the issue started with KN’s statement that we generate numbers from sets, apparently starting with the empty set. Logically this is ludicrous nonsense. If it were a Buddhist statement, it could be imagined to make a spiritual point, but hardly otherwise. You seem to have nothing to say about this, which was the actual issue all along.

  17. Erik,

    It doesn’t imply that intersections won’t work (or whatever it is you are trying to say there).

    I figured that would go over your head. What I’m saying is that your choice makes the definition of ‘intersection’ awkward and inelegant. The mathematicians made the smarter choice.

    The problem with the set theory is they so desperately want to call even non-sets sets (“empty sets”).

    It has nothing to do with desperation and everything to do with intelligence. They’re simply too smart to make the choice you did. They made a smarter choice because they foresaw the consequences.

    You didn’t, and so you blundered into a trap of your own making. Now you’re trying to blame your mistake on them, in true crackpot fashion.

  18. Never done talking past the issue. Such is keiths.

    Okay, let’s go with this one:

    keiths:
    What I’m saying is that your choice makes the definition of ‘intersection’ awkward and inelegant.

    Prove it.

  19. Moreover, the issue started with KN’s statement that we [can] generate numbers from sets, apparently starting with the empty set. Logically this is ludicrous nonsense.

    No, the idea of using set theory to generate numbers makes perfect sense. You’re bad at mathematical and philosophical thinking, so you have trouble grasping this. But that’s merely a sign of your own shortcomings, not of a problem at the foundations of mathematics.

    You seem to have nothing to say about this, which was the actual issue all along.

    Add poor reading comprehension to your list of shortcomings. I addressed KN’s statement directly:

    KN:

    This isn’t that hard. The empty set is a set that contains no members. A set that contains the empty set has one member. A set that contains the set that contains the empty set has two members. That’s how you generate the natural numbers from sets.

    [Emphasis added]

    KN,

    The sentence in bold is incorrect. That set contains only one member, because members of an element don’t count as members of the set.

    What you’re forgetting is that each subsequent natural number is defined as the union of itself with the set containing itself:

    n + 1 ≡ n ∪ {n}

  20. keiths:

    What I’m saying is that your choice makes the definition of ‘intersection’ awkward and inelegant.

    Erik:

    Prove it.

    You’ll do that for me when you try to match the elegance of this definition:

    For any two sets A and B,

    A ∩ B = {x : x ∈ A ∧ x ∈ B}

  21. Show why the formulation would not work under what you take to be my definition. Because the actual issue is this: You never understood my objection to the term and never wanted to understand. You only like to call it crackpot. Now you must show concretely what the alleged crackpotness consists of.

  22. @keiths
    The equation says nothing about an empty set, unless you are assuming that x can be zero. This assumption is not evident in the equation. Whether sets that have no common elements have an intersection at all is a point that can be logically contested. Either way, it does not affect the mathematical notation. Sets with no common elements have ∅ in common, whatever name you give to it.

    In arithmetic you cannot divide by zero. Why should you be able to in set theory? What have those smart mathematicians told you, keiths?

    You are definitely assuming too much about what I am outlawing. In axiomatic set theory, everything is a set, axiomatically, even when it’s not (i.e. has no elements in it). Reality (and logic) doesn’t work like this. In my life, I have found the law of non-contradiction to be a good law. What’s a good reason to go against it?

    Should elegance trump logic? Trump is preferable because he is more elegant than Hillary? What if Trump is empty inside and Hillary a can of worms, how do you pick elegance then?

  23. Erik: Should elegance trump logic? Trump is preferable because he is more elegant than Hillary? What if Trump is empty inside and Hillary a can of worms, how do you pick elegance then?

    Pure brilliance. hahaha

  24. Erik: Should elegance trump logic? Trump is preferable because he is more elegant than Hillary? What if Trump is empty inside and Hillary a can of worms, how do you pick elegance then?

    No need to pick between them anymore, I guess. Hillary recommended an airstrike on Syria several hours before Trump (who ridiculed Obama for suggesting any such thing a couple years ago) ordered one.

  25. Erik: The problem with the set theory is they so desperately want to call even non-sets sets (“empty sets”). What’s wrong with calling it zero or emptiness? It’s emptiness, nothing, so what’s the urge to call it a set?

    Why would you want a special name (“emptiness?”) for something that in all respects functions in the same way as any set, except you have a fancyful notion that it somehow isn’t any good.

    Moreover, the issue started with KN’s statement that we generate numbers from sets, apparently starting with the empty set. Logically this is ludicrous nonsense.

    Perhaps you can come up with an actual logic argument for that?

    For myself, I am more inclined to think that “logically this is ludicrous nonsense” is itself ludicrous nonsense.

  26. Erik: Because the actual issue is this: You never understood my objection to the term and never wanted to understand.

    Perhaps you should put that down as your failure to successfully communicate your objection.

  27. Neil Rickert: Why would you want a special name (“emptiness?”) for something that in all respects functions in the same way as any set…

    List the functions.

    I’ve seen so many rules for “nonempty sets” that I have always wondered why there’s such desperation to insist that the empty set is a set. If the empty set were not a set, all those rules would be for sets, plain and simple. Much more elegant.

  28. Erik: List the functions.

    The important functions are:

    Unions of sets;
    Intersections of sets;
    Subsets (A is a subset of B);
    Membership (is x a member of set y).

    And those work just as well for the empty set as for any other set.

  29. walto: No need to pick between them anymore, I guess.Hillary recommended an airstrike on Syria several hours before Trump (who ridiculed Obama for suggesting any such thing a couple years ago) ordered one.

    BTW, I wonder if Rand Paul, who played golf at Mar-a-Lago last weekend

    (a) Thought about crony capitalism every time he walked down a par-5 fairway;
    (b) Will let everyone know when the swamp gets drained; or
    (c) Is enjoying the irony of the fact that Obama wouldn’t attack Syria without the consent of Congress, but Trump (the isolationist!) doesn’t give a shit about Congress.

    #Libertariansaredumbshitdupes

  30. <blockquote cite="c

    Neil Rickert:

    Perhaps you can come up with an actual logic argument for that?

    For myself, I am more inclined to think that “logically this is ludicrous nonsense” is itself ludicrous nonsense.

    Erik thinks that a set is defined as having members, so that there cannot be a set without members any more than there can a square circle.

  31. Neil Rickert: The important functions are:

    Unions of sets;
    Intersections of sets;
    Subsets (A is a subset of B);
    Membership (is x a member of set y).

    And those work just as well for the empty set as for any other set.

    The standard non-answer.

    I’m not the first to come up with questions like this. What is clear is that nobody here is interested in the answers https://en.wikipedia.org/wiki/Empty_set#Philosophical_issues

    Seeing the empty set as philosophically suspect is not a new thing among philosophers. This statement works when we count KN as a non-philosopher.

  32. Neil, to Erik:

    Perhaps you should put that down as your failure to successfully communicate your objection.

    I understand his objection. The problem isn’t that he’s failed to communicate it, it’s that he’s failed to defend it.

    He just keeps yammering “But a set can’t be empty!”, as if mathematicians were obligated to use the same dumb definition of set that he uses.

  33. keiths: I understand his objection.

    No you don’t. You still don’t.

    keiths: He just keeps yammering “But a set can’t be empty!”, as if mathematicians were obligated to use the same dumb definition of set that he uses.

    And you just keep yammering that mathematicians are smart and I am a crackpot. As if I were obligated to accept whatever mathematicians say. Real mathematicians provide proofs.

  34. Erik: Seeing the empty set as philosophically suspect is not a new thing among philosophers.

    Seeing philosophy as philosophical suspect is not a new thing either.

    Much that is in that “philsophical issues” section would be considered the work of cranks (not to be confused with crackpots).

    However, one “issue” was this:

    “…was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object.”

    The same could be said about any mathematical entity.

    The alleged issues seem to come from wanting natural language to be logical. But natural language has never been logical, and a logical language could never serve as a natural language.

  35. Neil Rickert: The alleged issues seem to come from wanting natural language to be logical. But natural language has never been logical, and a logical language could never serve as a natural language.

    Nope. The conflict is in metaphysics. You don’t assign to an empty spot arbitrarily the value of a box or a house or a living or a dead entity. Not without a Very Good Reason anyway. Until then it’s empty, nothing more.

    Thus far, the only Very Good Reason has been provided by keiths – elegance. However, my proposal also has elegance going for it. When we call ∅ for example “null”, we will get rid of the unelegant concept “nonempty sets” which occurs in set theory far too often. “Nonempty sets” would become sets, plain and simple. And elegant.

  36. Erik,

    The equation says nothing about an empty set, unless you are assuming that x can be zero.

    It’s a definition, not an equation, and of course it says nothing explicitly about the empty set. That’s the whole frikkin’ point! It doesn’t have to say anything about the empty set because the definition works for all sets, empty or not.

    Whether sets that have no common elements have an intersection at all is a point that can be logically contested. Either way, it does not affect the mathematical notation.

    It does affect the mathematical notation. Mathematicians use curly braces to enclose the contents of a set. {a, b} is a set containing two elements. {x} is a set containing one element. { } is a set containing no elements — the empty set.

    The definition of ‘intersection’ uses curly braces:

    A ∩ B = {x : x ∈ A ∧ x ∈ B}

    If A and B have no elements in common, then the braces end up empty:

    A ∩ B = { }

    It’s perfectly coherent. It just requires you to use the mathematicians’ concept of set instead of the dumb Erik concept. One definition applies in all cases.

    If you use the Erik concept, then you can’t apply the elegant definition unless you already know that the sets have at least one member in common. That’s hopelessly kludgy and awkward.

    Mathematicians are smarter than that.

  37. Erik: The conflict is in metaphysics.

    All of mathematics should be seen as outside of metaphysics. If you are trying to relate mathematics to metaphysics, you are doing it wrongly.

  38. Erik:

    In arithmetic you cannot divide by zero. Why should you be able to in set theory?

    You can’t. The empty set isn’t analogous to dividing by zero, it’s analogous to zero itself. (And it is zero itself when set theory is used as the foundation of math, in the standard way.)

    In my life, I have found the law of non-contradiction to be a good law. What’s a good reason to go against it?

    You are the one creating the contradiction by insisting on your goofy definition of ‘set’. Mathematicians avoid the contradiction by using a better definition.

  39. Neil Rickert: All of mathematics should be seen as outside of metaphysics.If you are trying to relate mathematics to metaphysics, you are doing it wrongly.

    The same way as you think it justified to treat a non-set as if it were a set, I am justified to treat mathematics as if it were metaphysics. And my justification is better than yours because both mathematics and metaphysics essentially deal with logical structures, so mathematics is not a non-metaphysics.

    There, I just gave a good reason for what I’m talking about. Do you have a good reason for what you are talking about, besides that set theory has axioms and that you are an op here? Any standard for your “you are doing it wrongly”?

  40. keiths:
    A ∩ B = {x : x ∈ A ∧ x ∈ B}

    If A and B have no elements in common, then the braces end up empty:

    A ∩ B = { }

    It’s perfectly coherent.

    The two statements can be equated if x can be zero or empty and division with such is permitted. These assumptions are not explicitly seen in the first statement. Until then it’s not perfectly coherent. It’s just yammering and assertion.

    By the way, what are your thoughts on Gödel’s incompleteness theorems?

  41. Erik: Do you have a good reason for what you are talking about, besides that set theory has axioms and that you are an op here? Any standard for your “you are doing it wrongly”?

    From my viewpoint, mathematics is all about method. Existence has nothing to do with it. Numbers are just fill-ins (or fictions) that stand for whatever is being counted. Mathematics gets its generality from the fact that it isn’t about anything and therefore can potentially be used for everything.

    Back to sets. The set operations of intersection and union are analogous to the logical operations of disjunction and conjunction. If you want to make intersection have special cases by ruling out the empty set, then you ought to also require special cases for conjunction. Disallowing the empty set is analogous to disallowing FALSE in logic.

  42. Kantian Naturalist: Erik thinks that a set is defined as having members, so that there cannot be a set without members any more than there can a square circle.

    Since number expresses a quantity there cannot be a number zero. I think I’m getting it.

  43. This bruhaha is bit like arguing over whether zero is a natural number. If Erik wants to define “set” so that no set can be empty, he can do that. He’ll need different axioms to get arithmetic, but who the hell cares?

    I mean, when he insisted that there can never be laws allowing for marriages between two women or two men, that actually mattered. This, not so much. Maybe reduction to set theory would be less elegant, using his definitions, but I’m guessing an expert in mathematical logic could make do with Erik’s revisions.

    They’re pointless, but also harmless.

  44. walto,

    You’re not getting it.

    Erik is free to choose whatever clunky definitions and axioms he wants and to spin his wheels trying to get them to work elegantly. People have some strange hobbies.

    The kerfuffle is over

    a) his claim that the mathematical community has screwed up by adopting the supposedly self-contradictory notion of the empty set; and

    b) his naive grandiosity in thinking that he, a mathematically unskilled and undereducated philologist, has discovered a flaw in set theory that has evaded mathematicians since the days of Cantor.

    The latter is particularly crackpottish. Erik is fighting to avoid considering an obvious alternative that is thousands of times more likely: that he simply doesn’t understand what the mathematicians are up to.

    The problem lies with him.

  45. petrushka,

    Since number expresses a quantity there cannot be a number zero. I think I’m getting it.

    If Erik were a zero skeptic, we could at least admire his consistency. But he embraces zero and rejects the empty set. Go figure.

  46. Erik is simply withholding his logical proof of the self-contradiction in the definition of empty set until someone pledges allegiance to the law of non-contradiction, so here I go:

    “I pledge allegiance to the Law of Non-Contradiction, and to Plato’s Republic, for which it stands, one Logic under God, indivisible, with premises and conclusions for all.”

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